cds bought at a bank verses cd’s bought from a brokerage ... · annual percentage yield (apy)....

Bank vs. Brokered CDs Spring, 2011 1

CDs Bought at a Bank verses CD’s Bought from a Brokerage

Floyd Vest CDs bought at a bank. CD stands for Certificate of Deposit with the CD originating in a FDIC insured bank so that the CD is insured by the United States government. Consider the following “extra simple” example where the CD buyer has opted to reinvest the interest payments back into the CD at the CD rate. We made this example very simple to demonstrate interest paid on interest. Example 1 Assume a person can buy a two-year CD for $1000 that pays an annual nominal interest rate (APR, Rate) of 5%. Dividends are paid and compounded semiannually so a

(simulated) semiannual rate is 5%2

= 2.5% = 0.025.

How much is the Final Balance of the CD and what is the Annual Percentage Yield (APY, Yield)? Calculating the Final Balance. The semiannual dividends (interest payments) are 0.05

2(1000) = $25.

The following is a drawn out derivation, using simulated bank postings, illustrating earning interest on interest: Posting 1: Dividend 1 = 1000(0.025) Postings 2, 3, and 4 include accumulated interest on interest and the current dividend. The factor of (1 + 0.025) results from compounding. (See the Compound Interest Formula in a Side Bar Note.) Posting 2: 1000(0.025)(1 + .025) + 1000(0.025) Posting 3: 1000(0.025)(1 + 0.025 2) + 1000(0.025)(1 + 0.025) + 1000(0.025) Posting 4: 1000(0.025)(1 + 0.025 3) + 1000(0.025)(1 + 0.025 2) + 1000(0.025)(1 + 0.025) + 1000(0.025) + 1000, where $1000 is the price of the CD. Posting 4 gives the Final Balance including principal plus interest. We can calculate the Final Balance by factoring to get (1) 1000(0.025)[(1 + 0.025 3) + (1 + 0.025 2) + (1 + 0.025) + 1] + 1000 We recognize the expression in brackets as the sum of an ordinary annuity. (See Sum of an Ordinary Annuity in the Side Bar Notes. For the basic Mathematics of Finance formulas for Compound Interest and Annuities, see Luttman or Kasting in Unit 1 of this course.) The sum is

(2) (1+ 0.025)4!1

0.025 = 4.1525 .

Then we calculate to get 1000(0.025)(4.152) + 1000 = $1103.81 as the Final Balance of the CD. The bank will not quite do it this way as can be seen in an exercise. To


summarize, the CD has earned interest of $103.81 and the principal was repaid. The interest on interest was $3.81. (See the exercises and Side Bar Notes for formulas.) You Try It #1 (a) Do a derivation similar to the above Example 1 for a semiannual two-year bank CD of $10,000 that pays an annual 2% nominal rate. By this activity, you will be demonstrating earning interest on interest and deriving a formula for calculating the Final Balance. (b) Draw and label a time line. (c) Use the Formula for the Sum of an Ordinary Annuity given in a Side Bar Note to calculate the Final Balance. (d) How much interest was earned? How much interest on interest was earned? (e) Use a general Formula for the Sum of an Ordinary Annuity to give a general closed form formula for the Final Balance of a bank CD. Label all your variables. Annual Percentage Yield (APY). What is the APY that the bank will report for the CD in Example 1? By definition the APY is annual rate that pays the same as the periodic rate compounded for a year. Thus APY = (1 + 0.025 2) – 1 = 0.0506 = 5.06%. (See an explanation in the You Try Its.) To summarize, the 5% nominal rate compounded semiannually yielded actual annual earnings of 5.06% compounded annually; i.e., 1000(1 + 0.056 2) = $1103.81. Also, we have earned interest on interest. Typically, the bank will report the APY as a percent with two digits to the right of the decimal, and the annual nominal rate (APR, Rate) will be reported as a percent with three digits to the right of the decimal point. You Try It #2 (a) For the CD you calculated in You Try It #1, calculate the APY. (b) Give a general meaning of APY. If i is the periodic rate compounded k times per year, give two basic formulas for APY. The way banks accumulate interest in a CD. Consider a $150,000, nine-month CD paying interest monthly at the annual nominal rate of 2.440%. In the calculations, the bank carries the principal of $150,000 and accumulated interest forward each month to get a Balance brought Forward to the beginning of the month. Then, it calculates interest for the month on this balance and adds it to the Balance brought Forward to get the end of month Ending Balance. For example, on 8/01 the Balance brought Forward is $150,576.26. Then on 8/31 the

Deposit Dividend = 0.02440365

31

(150,576.26) = 312.04 .


The Deposit Dividend of 312.04 is added to the Balance brought Forward of 150,576.26 to get the Ending balance of $150,888.30 posted 8/31. Please note that the denominator is never equal to 12, the number of months in a year. You Try It #3 (a) Do a derivation of a formula for Final Balance by following the above steps from the beginning of the investment of principal P, in a two year semiannual CD, with an annual

nominal rate r, with the semiannual rate i = 2r . Start with the Beginning of the first six

months, then to the End of the first six months, etc. Factor and use exponents and label formulas at each step. You will end with a closed form formula for the Final Balance B. (b) Apply your formula to the CD in Formula 1 in the above discussion. You should get the same answer. Do you recognize the formula that you derived? You now have two formulas for calculating the Final Balance B of a bank CD. CDs bought from a broker. Brokered CDs originate in an FDIC Insured bank but are sold by a broker. Typically, new brokered CDs do not offer the option of reinvesting the dividends (interest) in the CD at the CD rate. Dividends are simply paid to the owner at the end of each dividend period. Brokered CD’s typically pay dividends semiannually. Calculating the Final Balance of a Brokered CD. In our famous example we would buy a $1000 two-year brokered CD paying dividends (interest) semiannually at the annual

nominal rate of 5%. Each dividend would be 0.052

(1000) = $25.

(Again, we are doing a simulation. See the You Try Its.) For Postings 1, 2, 3, and 4 for the CD, each would display a dividend of $25. Posting 4 would also display the price of the CD of $1000 as seen on the timeline in Figure 1. $25 $25 $25 $25 + $1000 ↑ ↑ ↑ ↑ __|__________|__________|___________|___________|_______________

0 12

1 112

2 years

↓ P = $1000 Fig. 1: A cash flow timeline for a brokered CD. The Value at Maturity would be $1000. Total interest earned is $100. The Final Balance is $1100. See the exercises and Side Bar Notes for other terms of a brokered CD.


Comparison of the two CD’s. The two CDs were both $1000 two year CDs with a nominal annual rate of 5%, and with dividends paid semiannually. For the CD bought at the bank with the option of reinvesting dividends at the CD rate, Final Balance = $1103.81. Total interest = $103.81. Interest on interest is $3.81. If you are not impressed with the $3.81, see the exercises. For the brokered CD, Value at Maturity = $1000. Total interest = $100. For the brokered CD, there is no assumption as to what happens in the future to the dividends. For this particular CD the Value at Maturity is the same as the price, but for CDs bought on the secondary market, this may not be true. The dividends of a brokered CD are typically paid into a “sweep” money market fund in the name of the CD owner. Of course, dividends might be reinvested at a “good” rate. (See the Side Bar Notes.) However, for the small investor, this is not easy to do and not convenient. They could spend the money, but the point may be to accumulate savings. The broker may charge an account maintenance fee but any commission on the CD is net the nominal interest rate; i.e., it is not subtracted from the announced interest payments or the Value at Maturity. Terminology for a brokered CD. The terms used for CDs by a brokerage are annual nominal rate (APR), Yield to Maturity (YTM), and Annual Percentage Rate (APR). The term Annual Percentage Yield (APY) is not used on this type of a brokered CD. You Try It # 4 Do the calculations, describe the events, draw a timeline, and use brokered CD terms, for a brokered CD similar to the bank CD in You Try It #1. The meaning of Yield to Maturity (YTM). By definition, YTM is the annual nominal rate that discounts the cash flows to the price of the CD. That is, in our case

(3) Price = 25 1+ YTM2

!"#

$%&

'1

+ 25 1+YTM

2

!"#

$%&

'2

+ 25 1+YTM

2

!"#

$%&

'3

+

25 1+YTM

2

!"#

$%&

'4

+1000 1+YTM

2

!"#

$%&

'4

.

We can try a YTM of 5% (our annual nominal rate) and see what it does. (4) Price = 25(1.025)-1 + 25(1.025)-2 + 25(1.025)-3 + 25(1.025)-4 + 1000(1.025)-4. Let’s multiply through by (1.025 4) to get (Price)(1.025)4 = 25[1.0253 + 1.0252 + 1.025 + 1] + 1000.


Observing that the expression in the brackets is the sum of an ordinary annuity, we have

(Price)(1.025)4 = 25 (1+ 0.025)4 !1

0.025

"

#$

%

&' + 1000 = 25(4.152) + 1000

Price = 4

25(4.152) 1000(1.025)

+ = $1000.

So the broker is likely to report that YTM for the brokered CD is 5%. The problem. The problem is that the expression in our YTM calculation of 25[1.0253 + 1.0252 + 1.025 + 1] indicates that interest is being paid on interest, compounded semiannually, at the nominal annual rate of 5%. The brokered CD didn’t do this. In finance books, calculator manuals, and other publications, the precaution is usually indicated that YTM assumes dividends are reinvested at the nominal annual rate of the CD compounded per period. Actually, the brokered CD paid 2.5% per six-month period with no indication of what happened to the $25 dividends. It’s just that people dealing with brokered CDs and bonds have to understand this issue. (See www.investopedia.com/terms/m/mirr.asp. Examining the YTM (Yield to Maturity) concept. To examine YTM, consider our example of semiannual payments, with YTM = 2i, where i is the interest rate per six-month period. Let P = price of the CD (bond, stock, security, business investment). Assume the following timeline for cash flows consisting of (dividends) interest payments I and value at maturity M, for n payment periods.

1I 2I 3I 4I 1nI − In+M

↑ ↑ ↑ ↑ . . . ↑ ↑ _| ______|______|______|______|____________|_______|________

↓ time P

Fig. 2: Cash flow time line for n periods. We will divide price P into n parts so that P = P1 + P2 + P3 + … + Pn. Then invest each Pj at the rate i compounded per period to earn the cash flows. This gives: Multiplying through by (1 + i)j gives: I1 = P1(1 + i) P

1= I

1(1+ i )!1

I2 = P2(1 + i)2 22 2 (1 )P I i −= +

I3 = P3(1 + i)3 P3= I

3(1+ i )!3

. . . . . . M + (1 )nn nI P i= + (1 ) (1 )n n

n nP I i M i− −= + + +


Remembering that P = P1 + P2 + P3 + … + Pn , we get (4) P = 1 2 3

1 2 3(1 ) (1 ) (1 ) ... ( )(1 ) nnI i I i I i I M i− − − −+ + + + + + + + + So YTM is the annual nominal rate that discounts the cash flows to the price P. We have derived here the formula for YTM from basic principles. If there are semiannual dividends, the YTM = 2i. Also remember that YTM assumes reinvestment of dividends at the annual nominal YTM rate. One can solve a YTM equation for i by multiplying through by (1 + i)n to get a polynomial . If the Is are equal, you can use the formula for the sum of an ordinary annuity to assist. But you still will want to use a computer program. If the Is are unequal, you simply use available calculators or computers to solve the nth degree polynomial for a root (zero) between 1 and 2, thus giving an i between 0 and 1. Using the Financial Functions in the TVM Solver on the TI83/84. We will consider a two-year $10,000 bank CD above, with an annual nominal rate of 5%, paying Dividends semiannually at the end of each period, and Dividend of $250. We will solve for i = 0.05/2. We will use the names of variables and the sign conventions of the TVM Formulas. In the TVM Solver we will enter 4 as N, (-)10000 as PV, 250 as PMT, and 10000 as FV, and calculate i. Code and comments: 2nd Finance Enter 4 Enter ∨ to PV (-)10000 Enter 250 Enter 10000 Enter For P/Y write 1. Enter Select Pmt:End ∧∧∧ to highlight I% Alpha Solve You read 2.50 So i = 2.5% = 0.025 and the annual nominal rate = 5% = 0.05. Due to the curious nature of the definition of YTM for a brokered CD, this will also work for a brokered CD. See the Exercises.


Exercises For all exercises, show all your work, label inputs, give formulas, label outputs, and summarize. 1. For the $1000 Brokered CD discussed above, use the TVM Solver on the TI 83/84 to calculate i (the semiannual rate). What is YTM? Substitute the numbers into an appropriate formula. (See Exercise 6.) There is no closed form formula for solving for i in an annuity. 2. Calculate the APY to compare the following five-year bank CDs. CD No.1 pays 2.3% nominal, compounded monthly. CD No. 2 pays 2.4% nominal compounded semiannually. 3. Use two formulas, one in You Try It #1, and the other in You Try It #3, to calculate the Final Balance for a five-year $4000 bank CD compounding monthly at a nominal rate of 6%. 4. Use the formulas for Final Balance B in You Try It #1 and in You Try It #3 to derive

the formula 0 = P + R(1+ i )n !1

i

"

#$

%

&' ! P (1+ i )

n . This is close to one of the TVM

formulas in the Appendix for the TI 83/84 that is used by the TVM Solver. Look it up and make the comparison in variables, Gi, and sign conventions. 5. Formula 4 above suggests the need for a closed form formula for the Sum (Present

Value) of these discounted values, R. Derive the formula A = R1! (1+ i )!n

i

"

#$

%

&' , which

does this. 6. Show from Formula 4 that for a brokered CD, a general formula for Price is

Price = R1! 1+ i( )

!n

i

"

#

$$

%

&

''+M 1+ i( )

!n, where R is the semiannual payment,

i = YTM/2. n = (the number of years) × 2. M is the Value at Maturity, which can be the same as Price. 7. Explain why the term APY is not appropriate for brokered CDs. 8. Consider a $200,000, five-year bank CD, at 10% nominal rate compounded monthly. (a) Calculate the monthly interest dividends. (b) Calculate the future value of the interest payments with interest reinvested at the CD rate. (c) Calculate the Final Balance of Principal plus Interest. (d) Calculate the total Dividends. (e) Calculate the interest on interest.


9. To summarize, give two formulas for the Final Balance B for a Bank CD where P is the initial deposit, r is the nominal annual rate (yield), n is the number of years, k is the number of compounding periods per year, and R is the periodic dividend. 10. To summarize, give a formula for the Final Balance B for a newly issued Brokered CD where Price P = Value at Maturity M, with semiannual dividends R, for n semiannual

periods, 2n years, an APR = YTM = nominal yearly rate, and R = M

2YTM⎛ ⎞

⎜ ⎟⎝ ⎠.

11. To summarize, give the formula for the Price P of a newly issued Brokered CD with

semiannual dividends R, n semiannual periods, 2n years, Value at Maturity M,

and i = 2

YTM as the semiannual rate where the annual nominal rate = YTM = APR


Side Bar Notes General Formula for the Sum of an Ordinary Annuity with payments R at the end of each

period, interest rate i per period, for n periods: (1 ) 1niS Ri

⎡ ⎤+ −= ⎢ ⎥⎣ ⎦

.

Compound Interest Formula for the sum S, from investment of Principal P, invested for n periods, with i the interest rate per period: S = P(1 + i)n. Shopping for CD rates. The following are instructions for using the Internet to shop for and compare CD rates from banks and brokerages.

• For CDs bought from a bank: go to www.bankrate.com. Click on the Find a CD or MM rate box. Select the terms you want. Click National. You can do business nationally just as easily as locally. Click on Sort, and APY to sort by APY. Write down the 1-, 2-, and 5-year rates. Write down the National Average rates and compare with the top rates. Do some math to compare the differences on a $100,000 CD from age 30 to 70.

• For CDs bought from a brokerage, go to www.vanguard.com. Click on Go to Personal Investors Site. Under Research Funds and Stocks, click Stocks, Bonds, and CDs. Chose your terms and jot down the rates. Do some more math and comparisons. Summarize your observations and discuss with your teacher and your class. At one time, CD rates were as high as 14%. What is difference in earnings between 14% and 12% CDs over time? Why so much difference?

“Don’t get trapped in the rate race.” This was advice published in large print on the cover of a brochure by a local Credit Union. Do you think it is good advice? FDIC insured bank and brokered CDs. See http://www.fdic.gov/deposit/deposits/insured/. Interest paid by brokered CDs. Brokered CDs typically pay interest on the Principal P at

the YTM rate by the formula Dividends = YTM365

x

!

"

###

$

%

&&&P( ) , where x is the number of days in

the six month period. You notice that the denominator is seldom 2, except possibly during leap year.


Modified Internal Rate of Return (MIRR). The basic definition of YTM is the same as that of Internal Rate of Return (IRR). IRR has the same problems as YTM. For brokered CDs¸ one problem is the assumption that dividends are reinvested at the CD rate. To compensate, modifications of IRR (YTM) such as Modified Internal Rate of Return (MIRR) are used. Spreadsheets have built in functions to calculate MIRR. If you use MIRR, you will be asked for a “reinvestment rate” for positive cash flows and a “finance rate” for negative cash flows. Consider the following $1000, two-year brokered CD with YTM = 5% and paying dividend of $25 at the end of every six months. Consider reinvestment of dividends at 4% YTM semiannually up to the maturity date of the CD. What is the MIRR? See http://en.wikipedia.org/Modified_internal_rate_of_return. “APR and APY: Why Your Bank Hopes You Can’t Tell The Difference” . See http://investopedia.com/articles/basics/04/102904.asp.


Answers to You Try Its. You Try It #1

(a) (b) (c) Final Balance = 100 (1+ 0.01)4 !1

0.01

"

#$

%

&' + 10,000 = $10,406.04.

(d) Total interest = 10,406.04 – 10,000 = $406.04 Interest on interest = $6.04 (e) One version of a general formula for the Final Balance B for a bank CD is the

following: B = R (1 ) 1nii

⎡ ⎤+ −⎢ ⎥⎣ ⎦

+ P where R is the Dividend payment, i is the interest rate

per Dividend period, n is the number of compounding periods (Dividend periods), and P is the Price of the CD. You Try It #2 (a) 1 + APY = (1 + 0.01)4, APY = (1 + 0.01)4 – 1 = 0.0201 = 2.01% (b) The APY is the annual interest rate that yields the same as the periodic rate compounded for a year. If i is the periodic rate compounded k times per year, then APY is such that (1 + APY) = (1 + i)k and APY = (1 + i)k – 1. The APY (sometimes called Effective rate) is used to compare the rates of different investments including different bank CDs. You Try It #3 (a) You should get the formula for the Final Balance B = P(1 + i)4, where i = r/2 and r is the annual nominal rate, and P is the principal invested in the CD. In general, if i is compounded k times per year, then i = r/k. If n is the number of years, then

B = P(1 + rk)nk . Exercise: From this formula, give a formula for P the price of a CD.

Solve for rk

. Solve for nk.

(b) B =1000 1+0.05

2

!"#

$%&4

= $1103.81. You have derived a version of the Compound

Interest Formula. You Try It #4

For this brokered CD, i = r/2 = YTM2

= 0.02/2 = 0.01 is the six month rate.

M = $10,000 = Price = Value at Maturity. Dividend = R = 0.01(10,000) = $100 per six months. Final Balance = 4R + M = 4(100) + 10,000 = $10,400. YTM = r = 2%.


Answers to Exercises 1. i = 2.5%. YTM = 5%.

2. APY for No. 1 = 1+ 0.02312

!"#

$%&12

'1 = 0.023244.

APY for No. 2 = 1+ 0.02412

!"#

$%&2

'1 = 0.024144. No. 2 pays a better rate.

3. CD for 5 years, 6% compounded monthly, P = $4000, i = 0.06/12 = 0.005. R = 0.005(4000) = $20. B is the Final Balance. Substituting in one formula gives

B = 20 (1+ 0.005)60 !1

0.005

"

#$

%

&' + 4000 = $5395.40.

Substituting into another formula gives B = 4000(1 + 0.005)60 = $5395.40. 6. From the cash flow timeline in Figure 1, and from Formula 4 and Exercise 5, the

present value of the Rs is 1 (1 ) niRi

−⎡ ⎤− +⎢ ⎥⎣ ⎦

, and the present value of M is M(1 + i)-n.

So the price of the brokered CD is P = 1 (1 ) niRi

−⎡ ⎤− +⎢ ⎥⎣ ⎦

+ M(1 + i)-n.

8. (a) Monthly dividends = 0.1012

(200,000) = $1666.67. With a bank CD, the customer

has the option of taking the dividends or reinvesting them in the CD at the CD rate. (b) Future Value of interest payments = $129,062.04. (c) Final Balance = 129,062.04 + 200,000 = $329,062.04 (d) Total of the dividends = 60 × 1666.67 = 100,000. (e) Total interest on interest = $29,062.04. 9. For a Bank CD, for the variables defined in the problem, where P is the initial deposit, r is the nominal annual rate (yield), n is the number of years, k is the number of compounding periods per year, and R is the periodic dividend,

Final Balance B = P 1nkr

k⎛ ⎞+⎜ ⎟⎝ ⎠

. Also B = R

1+r

k

!"#

$%&

nk

'1

r

k

(

)

*****

+

,

-----

+ P


10. For a newly issued Brokered CD, for the variables defined in the problem, where Price P = Value at Maturity M, with semiannual dividends R, for n semiannual periods,

2n years, and APR = YTM = nominal yearly rate, and R = M

2YTM⎛ ⎞

⎜ ⎟⎝ ⎠,

Final Balance B = nR + M = nM2

YTM M⎛ ⎞+⎜ ⎟⎝ ⎠.

11. For a newly issued Brokered CD, for the variables defined in the problem, where i is

the semiannual rate = 2

YTM , where the annual nominal rate = YTM = APR for n

semiannual periods for2n years , with semiannual dividends R, and Value at Maturity M,

Price P = 1 (1 ) (1 )n

niR M ii

−−⎡ ⎤− + + +⎢ ⎥

⎣ ⎦.


Teachers’ Notes Shopping for CD rates. One of the Side Bar Notes gives instructions for shopping and comparing the rates paid by brokered CDs and CDs bought at a bank, and shopping for top rates paid nationally. You might have some students report their research in this area, and pass out copies of their reports. Retirement savings are often done in IRAs, 401ks, and 403bs with an insurance company or a mutual fund family. Some will have a “brokerage window” where you can invest in CDs. At times CDs pay better rates than fixed accounts and money market funds. At a brokerage window, you can also invest in individual bonds and stocks. Buying and holding a collection of bonds to maturity can at times pay better than a fixed account or a bond fund. The advantage of owning your own bonds over a bond fund is you know their worth at maturity while the NAVs of a bond fund vary unexpectedly with market interest rates. See articles in this course on bonds including interest rate risks and bond duration. A lifetime file on personal finance. Have your students set up and maintain a lifetime file that includes articles from this course. Hopefully, some day, in the near future, they will use them to manage successfully their own finances. For example, this article gives some good web sites. There are hundreds of advantages to knowing financial mathematics and personal finance. Have your students keep an accumulating list of the useful things they learned in this course. As you teach this course, you can accumulate items for pre- and post-tests and questionnaires on financial knowledge, attitudes, and personal values, and mathematics knowledge and skills. Have students use this list to evaluate the course at the end. The basic formulas for Bank and Brokered CD’s are summarized in the answers to Exercises 9, 10, and 11.

cds bought at a bank verses cd’s bought from a brokerage ... · annual percentage yield (apy)....

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